2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/170554We study the local holomorphic Euler characteristic $χ(x,\mathcal{F})$ of sheaves near a surface singularity obtained from contracting a line $\ell$ inside a smooth surface $Z$. We prove non-existence of sheaves with certain prescribed numerical invariants. Non-existence of instantons on $Z$ with certain charges follows, and we conclude that $\ell^2$ poses an obstruction to instanton decay. A Macaulay 2 algorithm to compute $χ$ is made available at http://www.maths.ed.ac.uk/~s0571100/Instanton/15 pages. Revision history: v1: Original. v2: minor corrections, as publishedAlgebraic GeometryHigh Energy Physics - Theory14J60, 14J81text